In another question a comment mentioned that the change in government in the Ukraine conformed to UN norms.
Do such UN norms for government changes (e.g. impeachment of the president) exist?
If so, what are they?
There are not any specific norm or norms for “change of government” which may mean whether a change of government can be assumed internationally legal or illegal. This is a political subject-matter, and therefore, there are certain related issues that can be raised here. These issues will clarify the answer and therefore it is necessary to pay attention to them:
If we try to summary the whole answer, then we may say that the norms are not there, but two important aspects have to be considered: Threat to peace and security and massive human rights violations; meanwhile, if there are not such cases, and the change has gone through the due process of law and the new government has been able to gain the most support of the member states, then the new government is recognized. In this case, one may say that the government has changed (or has been changed) based on UN norms.
I think the question is about when the new regime is recognized by the UN and when it is not. In fact most coups and revolutions happening in the world pass without any troubles for the country's status in the UN, but there are exeptions
In practice this mostly depends on the position of the diplomats representing the state in the UN and their superiors, that is the minister of foreign affairs of the country in question.
If the diplomats recognize or support the government change, are willing to represent the new regime or give up their powers to those appointed by the new regime, then the representation of the country in the UN passes to the new regime without much troubles.
If on the other hand, the diplomats are unwilling to represent the new regime and maintain that they are only subordinated to the former head of state, then there are much troubles, and a UN general assembly resolution is usually required.
In the case of Ukraine, the former prime minister, Arbuzov, peacefully resigned and declared he is willing to cooperate with the new head of state. As such, the appointment of the new prime minister, Yatsenuk by the disputed acting president was accepted by the apparatus of the cabinet of ministers without a conflict (and yes! the final word belonged to a girl from the personnel department, who decided whether to accept the order signed by Turchinov!). So was replaced the foreign minister.
The majority of the diplomats after initial disorientation accepted the lead of the new minister. So did the representatives in the UN, and as such, the coup in the Ukraine passed without much troubles in the UN.
As another example you may consider the consequences of the Japanese occupation of Thailand in World War II. In December, 1941 Japan invaded Thailand and installed occupation regime in this country. Although the leading figures in the cabinet were kept unchanged, the supreme authority in the Thai politics now on belonged to Japan with the Thai government remaining as a puppet state. Under Japanese supervision the Thai government declared war on both the US and the British. The declaration of war was not signed by the regent though which was a violation of the procedure.
Given these circumstances, the Thai diplomats in Britain and the US acted differently. The ambassador in the UK delivered the war declaration to the British authorities, while the ambassador in the US considered the declaration issued under occupation and in violation of the procedure void.
As such, the US did not consider itself in the state of war with Thailand throughout the war while the British did and demanded reparations after the war concluded.